A discrete multivariate mean value theorem with applications

Talman, Dolf; Yang, Zaifu

doi:10.1016/j.ejor.2007.09.036

European Journal of Operational Research

2009

DOI: 10.1016/j.ejor.2007.09.036

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A discrete multivariate mean value theorem with applications

Dolf Talman

Zaifu Yang

Abstract: We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions.

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Cited by 6 publications

References 29 publications

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Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Laan

Talman

Yang

2009

J Optim Theory Appl

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Tucker's well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, . . . , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1, ±2, . . . , ±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n + q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

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Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Laan

Talman

Yang

2009

J Optim Theory Appl

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Discrete fixed point analysis and its applications

Yang

2009

J. Fixed Point Theory Appl.

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We formulate and prove several general theorems on existence of fixed (and zero) points for mappings defined on discrete sets in finite-dimensional Euclidean spaces. One of the major conditions imposed on the mappings under consideration is that of local gross direction preservation. In addition, economic applications are discussed. (2000). 47H10, 54H25, 55M20, 90C33, 91B50. Mathematics Subject Classification

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Solving discrete systems of nonlinear equations

Laan¹,

Talman²,

Yang³

2011

European Journal of Operational Research

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In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z n of the n-dimensional Euclidean space IR n . It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Z n and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the 'continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.

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A discrete multivariate mean value theorem with applications

Cited by 6 publications

References 29 publications

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Discrete fixed point analysis and its applications

Solving discrete systems of nonlinear equations

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